Operations with Rational Numbers: Grade 7 Study Guide
Grade 7 completes the rules for all four operations on rational numbers, positive and negative.
A rational number is any number that can be written as a fraction a/b with b ≠ 0. This includes integers, fractions, terminating decimals and repeating decimals. Every rational number has an opposite (additive inverse): the sum of a number and its opposite is 0.
Adding a negative number moves left on a number line, and subtracting a number is the same as adding its opposite: 3 − (−5) = 3 + 5 = 8.
Sign rules
| Operation | Rule | Example |
|---|---|---|
| Add, same signs | add, keep sign | (−4) + (−3) = −7 |
| Add, different signs | subtract, sign of larger absolute value | (−9) + 5 = −4 |
| Subtract | add the opposite | 2 − 7 = 2 + (−7) = −5 |
| Multiply or divide, same signs | positive | (−6) × (−2) = 12 |
| Multiply or divide, different signs | negative | (−20) ÷ 4 = −5 |
Distance and absolute value
The distance between two numbers on a number line is the absolute value of their difference: between −3 and 5 it is |5 − (−3)| = 8. Real-life uses: temperatures, elevation, bank balances and sports yardage.
Fractions and decimals
To add −2/3 + 1/2, use a common denominator: −4/6 + 3/6 = −1/6. To multiply (−3/4) × (2/9) = −6/36 = −1/6. To divide (−5/8) ÷ (1/4) = (−5/8) × 4 = −5/2. A fraction becomes a decimal by long division, and the decimal either stops or repeats: 3/8 = 0.375 and 2/3 = 0.666…
Use the rules.
Calculate.
Positive or negative.
Is it near what you expected?
Worked examples and common mistakes
Example: A diver at −15 ft rises 6 ft, then dives 10 ft: −15 + 6 − 10 = −19 ft. Common mistakes: dropping a negative sign, thinking −3 − 2 equals −1 and thinking that subtracting always makes numbers smaller. Draw a number line for every sign question until it feels automatic.
Check your understanding: model answers
1. 3. 2. 6/12 = 1/2. 3. 11.
✨ Key Takeaways
- Subtracting is adding the opposite.
- Like signs multiply to a positive.
- Distance = |difference|.
- Rational numbers can be written as terminating or repeating decimals.
💬 Questions for Discussion & Reflection
- Find (−8) + 11.
- Find (−3/4) × (−2/3).
- Find 7 − (−4).